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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Zhenfeng Liang"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# MTH9879 Homework 8\n",
      "\n",
      "Assigned: April 14, 2015.\n",
      "Due: April 28, 2015 by 6pm. \n",
      "\n",
      "Late homework **will not be accepted**.\n",
      "\n",
      "$$\n",
      "\\newcommand{\\supp}{\\mathrm{supp}}\n",
      "\\newcommand{\\E}{\\mathbb{E}}\n",
      "\\newcommand{\\Eof}[1]{\\mathbb{E}\\left[ #1 \\right]}\n",
      "\\def\\Cov{{ \\mbox{Cov} }}\n",
      "\\def\\Var{{ \\mbox{Var} }}\n",
      "\\newcommand{\\1}{\\mathbf{1} }\n",
      "\\newcommand{\\PP}{\\mathbb{P} }\n",
      "%\\newcommand{\\Pr}{\\mathrm{Pr} }\n",
      "\\newcommand{\\QQ}{\\mathbb{Q} }\n",
      "\\newcommand{\\RR}{\\mathbb{R} }\n",
      "\\newcommand{\\DD}{\\mathbb{D} }\n",
      "\\newcommand{\\HH}{\\mathbb{H} }\n",
      "\\newcommand{\\spn}{\\mathrm{span} }\n",
      "\\newcommand{\\cov}{\\mathrm{cov} }\n",
      "\\newcommand{\\sgn}{\\mathrm{sgn} }\n",
      "\\newcommand{\\HS}{\\mathcal{L}_{\\mathrm{HS}} }\n",
      "%\\newcommand{\\HS}{\\mathrm{HS} }\n",
      "\\newcommand{\\trace}{\\mathrm{trace} }\n",
      "\\newcommand{\\LL}{\\mathcal{L} }\n",
      "%\\newcommand{\\LL}{\\mathrm{L} }\n",
      "\\newcommand{\\s}{\\mathcal{S} }\n",
      "\\newcommand{\\ee}{\\mathcal{E} }\n",
      "\\newcommand{\\ff}{\\mathcal{F} }\n",
      "\\newcommand{\\hh}{\\mathcal{H} }\n",
      "\\newcommand{\\bb}{\\mathcal{B} }\n",
      "\\newcommand{\\dd}{\\mathcal{D} }\n",
      "\\newcommand{\\g}{\\mathcal{G} }\n",
      "\\newcommand{\\p}{\\partial}\n",
      "\\newcommand{\\half}{\\frac{1}{2} }\n",
      "\\newcommand{\\T}{\\mathcal{T} }\n",
      "\\newcommand{\\bi}{\\begin{itemize}}\n",
      "\\newcommand{\\ei}{\\end{itemize}}\n",
      "\\newcommand{\\beq}{\\begin{equation}}\n",
      "\\newcommand{\\eeq}{\\end{equation}}\n",
      "\\newcommand{\\beas}{\\begin{eqnarray*}}\n",
      "\\newcommand{\\eeas}{\\end{eqnarray*}}\n",
      "\\newcommand{\\cO}{\\mathcal{O}}\n",
      "\\newcommand{\\cF}{\\mathcal{F}}\n",
      "\\newcommand{\\cL}{\\mathcal{L}}\n",
      "\\newcommand{\\BS}{\\text{BS}}\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<font color = \"red\">Homework is to be done by each student individually.  To receive full credit, you must email a completed copy of this iPython notebook to Yu Gan (yugan323@gmail.com), Fubo Shi (fubo.shi.baruch@gmail.com), and Tai-Ho Wang (tai-ho.wang@baruch.cuny.edu) by the due date and time.  All R-code must run correctly and solutions must be written up neatly in Markdown/LaTeX format.\n",
      "\n",
      "<font color=\"blue\">If you encounter problems with Markdown/LaTeX or iPython notebook, please contact your TAs Yu Gan and/or Fubo Shi.\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### 0. (0 points) \n",
      "Read Section 6.7 of Bouchaud, Farmer and Lillo."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Predictability of order flow"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### 1. (6 points)\n",
      "\n",
      "Suppose that the distribution of meta order sizes $Q$ is exponential so that\n",
      "\n",
      "$$\n",
      "\\Pr(Q>V)= \\frac{1}{\\lambda}\\,e^{-\\lambda\\,V}\n",
      "$$\n",
      "\n",
      "Assuming slices of  constant size $\\bar v$, compute the conditional probability that a hidden order has more slices given that $n$ slices have already been observed.  Compare your result with the case where distribution of metaorder size is power-law."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Solution\n",
      "\n",
      "Denote $q_t$ is the conditional probability,\n",
      "\\begin{eqnarray}\n",
      "q_t&=&\\Pr(Q \\geq (n+1)\\bar v|Q \\ge n \\bar v) = \\frac{\\Pr(Q \\ge (n+1)\\bar v)}{\\Pr(Q \\ge n \\bar v)} = e^{-\\lambda \\bar v}\n",
      "\\end{eqnarray}\n",
      "\n",
      "As we can see from the above formula, the conditional probability is constant.\n",
      "\n",
      "Compare to the result with the case where distribution of metaorder size is power-law,\n",
      "\n",
      "\\begin{eqnarray}\n",
      "q_t&=&\\Pr(Q \\geq (n+1) \\bar v|Q \\ge n\\bar v) = \\frac{\\Pr(Q \\ge (n+1)\\bar v)}{\\Pr(Q \\ge n\\bar v)} \\sim \\left(\\frac{n}{n+1}\\right)^\\beta\n",
      "\\end{eqnarray}\n",
      "\n",
      "which increses with the order slices. The longer that the metaorder remains active, the larger the probability that it will continues."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Price manipulation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### 2. (10 points)\n",
      "\n",
      "Consider the price process\n",
      "\n",
      "$$\n",
      "S_t=S_0+\\int_0^t\\,f(\\dot{x}_s)\\,G(t-s)\\,ds+\\int_0^t\\,\\sigma\\,dZ_s\n",
      "$$\n",
      "\n",
      "where ${\\dot x}_s$ is the rate of trading in dollars at time $s < t$, $f(\\dot{x}_s)$ represents the impact of trading at time $s$ and $G(t-s)$ is a decay factor.\n",
      "\n",
      "(a) With the choices\n",
      "\n",
      "$$\n",
      "f(v)=\\sqrt{v},\\quad G(\\tau)=\\frac{1}{1+\\tau},\n",
      "$$\n",
      "    \n",
      "repeat the computations in Lecture 8  to derive an expression for the expected cost of accumulating shares at the rate $v_1$ and liquidating them at the rate $v_2$.\n",
      "\n",
      "(b) Verify that with $T=1$, $v_1=1/4$, $v_2=1$, the expected cost of this round-trip strategy is negative, thus demonstrating that price manipulation is possible. (Recall from the lecture slides that price manipulation is always possible if $f(\\cdot)$ is nonlinear and the decay kernel $G(\\cdot)$  is finite at the origin.)\n",
      "\n",
      "Hint: You might consider using Mathematica or Wolfram Alpha to do the integrations."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Solution\n",
      "\n",
      "### (a)\n",
      "\n",
      "Equation (7) from the note,\n",
      "\n",
      "\\begin{eqnarray*}\n",
      "C_{11}&=&v_1\\,f\\left(v_1\\right)\\,\\int_0^{\\theta\\,T}\\,dt\\,\\int_0^t\\,G(t-s)\\,ds \\nonumber\\\\\n",
      "C_{22}&=&v_2\\,f\\left(v_2\\right)\\,\\int_{\\theta\\,T}^T\\,dt\\,\\int_{\\theta\\,T}^t\\,G(t-s)\\,ds\\nonumber\\\\\n",
      "C_{12}&=&v_2\\,f\\left(v_1\\right)\\,\\int_{\\theta\\,T}^T\\,dt\\,\\int_0^{\\theta\\,T}\\,G(t-s)\\,ds\n",
      "\\end{eqnarray*}\n",
      "\n",
      "where $\\theta$ is such that $v_1\\,\\theta\\,T-v_2\\,(T-\\theta\\,T)=0$ so\n",
      "$$\\theta= \\frac{v_2}{v_1+v_2}$$\n",
      "\n",
      "In our case,\n",
      "\n",
      "$$\n",
      "f(v)=\\sqrt{v},\\quad G(\\tau)=\\frac{1}{1+\\tau},\n",
      "$$\n",
      "\n",
      "So, \n",
      "\\begin{eqnarray*}\n",
      "C_{11}&=&v_1\\,f\\left(v_1\\right)\\,\\int_0^{\\theta\\,T}\\,dt\\,\\int_0^t\\,\\frac{1}{1+t-s}\\,ds \\\\\n",
      "C_{22}&=&v_2\\,f\\left(v_2\\right)\\,\\int_{\\theta\\,T}^T\\,dt\\,\\int_{\\theta\\,T}^t\\,\\frac{1}{1+t-s}\\,ds\\nonumber\\\\\n",
      "C_{12}&=&v_2\\,f\\left(v_1\\right)\\,\\int_{\\theta\\,T}^T\\,dt\\,\\int_0^{\\theta\\,T}\\,\\frac{1}{1+t-s}\\,ds\n",
      "\\end{eqnarray*}\n",
      "\n",
      "With some calculation,\n",
      "\\begin{eqnarray*}\n",
      "C_{11}&=& v_1\\sqrt{v_1}((1+\\theta T)ln(1+\\theta T) - \\theta T)\\nonumber\\\\\n",
      "C_{22}&=& v_2\\sqrt{v_2}((1 + (1-\\theta)T)ln(1 + (1 -\\theta)T) - (1 - \\theta)T)\\nonumber\\\\\n",
      "C_{12}&=& v_2\\sqrt{v_1}((1+T)ln(1+T) - (1 + \\theta T)ln(1 + \\theta T) - (1 + (1-\\theta)T)ln(1 + (1 -\\theta) T))\n",
      "\\end{eqnarray*}\n",
      "\n",
      "So, the total expected cost is,\n",
      "\n",
      "\\begin{eqnarray*}\n",
      "C_{11} + && C_{22} - C_{12} \\\\\n",
      "&& = v_1\\sqrt{v_1}((1+\\theta T)ln(1+\\theta T) - \\theta T) \\\\\n",
      "&& + v_2\\sqrt{v_2}((1 + (1-\\theta)T)ln(1 + (1 -\\theta)T) - (1 - \\theta)T)\\\\\n",
      "&& - v_2\\sqrt{v_1}((1+T)ln(1+T) - (1 + \\theta T)ln(1 + \\theta T) - (1 + (1-\\theta)T)ln(1 + (1 -\\theta) T))\n",
      "\\end{eqnarray*}\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### (b)\n",
      "\n",
      "When $T = 1, v_1 = \\frac{1}{4}, v_2 = 1$\n",
      "$$\n",
      "\\theta = \\frac{v_2}{v_1 + v_2} = 0.8\n",
      "$$\n",
      "\n",
      "We plug into the above formula, we have,\n",
      "\n",
      "$$\n",
      "C_{11} + C_{22} - C{12} = -0.00370838\n",
      "$$\n",
      "\n",
      "We can see that the trading cost is negative, so the price manipulation is possible."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Set up R environment"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%load_ext rmagic"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The rmagic extension is already loaded. To reload it, use:\n",
        "  %reload_ext rmagic\n"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## The Obizhaeva-Wang model"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### 3. (8 points)\n",
      "\n",
      "In the Obizhaeva-Wang (OW) model, the price follows\n",
      "\n",
      "$$\n",
      "S_t=S_0+\\eta\\,\\int_0^t\\,{\\dot x}_s\\,e^{-\\rho\\,(t-s)}\\,ds+\\int_0^t\\,\\sigma\\,dZ_s\n",
      "$$\n",
      "\n",
      "(a) Derive an expression for the expected price path corresponding to a VWAP execution.\n",
      "\n",
      "(b) With the duration $T$ of the meta order set to 1, plot the expected price path from time $t=0$ to time $t=2$ for $\\rho=0.5,1,$ and $2$.\n",
      "\n",
      "(c) What is the expected average cost per share $\\bar S_T$ of a VWAP order in the OW model?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Solution\n",
      "\n",
      "### (a)\n",
      "\n",
      "Denote $n$ as the size of the metaorder, corresponding to a VWAP execution with duration $T$, $\\dot x = \\frac{n}{T}$\n",
      "\n",
      "Since the expectation of the noise term is zero, the expected price path is,\n",
      "\n",
      "\\begin{eqnarray*}\n",
      "E[S_t] &=& S_0 + \\eta \\int_0^{t \\wedge T} \\frac{n}{T}e^{- \\rho\\,(t - s)}\\,ds \\\\\n",
      "&=& S_0 + \\frac{n\\,\\eta}{\\rho\\,T}e^{-\\rho\\,t}(e^{\\rho\\,(t\\,\\wedge\\,T)}-1)\n",
      "\\end{eqnarray*}"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### (b)\n",
      "\n",
      "Without loss of generality, we set $S_0 = 0$, $n\\eta=1$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "\n",
      "T <- 1\n",
      "S_0 <- 0\n",
      "n_eta <- 1\n",
      "\n",
      "price_path <- function(t,rho){\n",
      "    \n",
      "    temp <- 0\n",
      "    if(t < T){\n",
      "        temp <- exp(rho * t)\n",
      "    }\n",
      "    else{\n",
      "        temp <- exp(rho * T)\n",
      "    }\n",
      "    \n",
      "    result <- S_0 + n_eta/(rho*T)*exp(-rho*t)*(temp - 1)\n",
      "    \n",
      "    return (result)\n",
      "}\n",
      "\n",
      "# Create a path vector\n",
      "t <- seq(from<-0, to<-2, by<-0.01)\n",
      "\n",
      "rho <- 0.5\n",
      "S_t <- sapply(t, function(x) price_path(x,rho))\n",
      "plot(t,S_t,col=\"red\",type=\"l\", main = expression(\"Expected price path\"))\n",
      "\n",
      "rho <- 1\n",
      "S_t <- sapply(t, function(x) price_path(x,rho))\n",
      "lines(t,S_t, col=\"blue\")\n",
      "    \n",
      "rho <- 2\n",
      "S_t <- sapply(t, function(x) price_path(x,rho))\n",
      "lines(t,S_t, col=\"green\")\n",
      "    \n",
      "legend(\"topright\", c(\"rho = 0.5\",\"rho = 1\",\"rho = 2\"),\n",
      "           lty = c(1,1,1), col=c(\"red\",\"blue\",\"green\"))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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f38DAQENDIy8vLyMjIycnh86gZdDu3ejZE+3aSa6HP/6AnR3CwzkaXd20KTw8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\nAElFTkSuQmCC\n"
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### (c)\n",
      "\n",
      "Under VWAP execution, $\\dot x = \\frac{n}{T}$\n",
      "\n",
      "The expected total cost is,\n",
      "\n",
      "\\begin{eqnarray*}\n",
      "C[\\Pi]&=&\\eta\\,\\int_0^T\\,{\\dot x}_t\\,dt\\,\\int_0^t\\,{\\dot x}_s\\,\\exp\\left\\{-\\rho\\,(t-s)\\right\\}\\,ds \\\\\n",
      "&=& \\eta\\,(\\frac{n}{T})^2\\,\\int_0^T\\,dt\\,\\int_0^t\\,e^{-\\rho\\,(t-s)}\\,ds \\\\\n",
      "&=& \\eta\\,(\\frac{n}{T})^2\\,\\frac{1}{\\rho}\\,\\int_0^T\\,1-e^{-\\rho\\,t}dt \\\\\n",
      "&=& \\frac{\\eta\\,n^2}{T^2\\,\\rho^2}(\\rho\\,T\\,+\\,e^{-\\rho\\,T}\\,-\\,1)\n",
      "\\end{eqnarray*}\n",
      "\n",
      "So, the expected average trading cost per share is,\n",
      "\n",
      "$$\n",
      "\\frac{C[\\Pi]}{n} = \\frac{\\eta\\,n}{T^2\\,\\rho^2}(\\rho\\,T\\,+\\,e^{-\\rho\\,T}\\,-\\,1)\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}